Proof of the Binomial Identity:

We aim to prove the following identity using a combinatorial argument:

$\binom{n}{k} = k \cdot \binom{n-1}{k-1}$

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Step 1: Left-hand side interpretation

The left-hand side, $\binom{n}{k}$, counts the number of ways to choose a subset of $k$ elements from a set of $n$ elements.

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Step 2: Right-hand side interpretation

The right-hand side, $k \cdot \binom{n-1}{k-1}$, can be interpreted as follows:

1. $\binom{n-1}{k-1}$: This counts the number of ways to choose a subset of $k-1$ elements from a set of $n-1$ elements.
2. $k:$ For each subset of $k-1$ elements chosen from the $n-1$, we pick one additional element from the remaining $k$-th element to make the subset of size $k$. Since there are $k$ different positions for this extra element in the final subset of size $k$, we multiply by $k$.

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Step 3: Combinatorial argument

1. Total count of subsets with $k$ elements: The left-hand side, $\binom{n}{k}$, directly counts all subsets of $k$ elements from $n$ total elements.

2. Breaking it down:

   • Start by picking one of the $k$ elements from the subset. Call this element $x$.
   • Now, the remaining $k-1$ elements of the subset are chosen from the other $n-1$ elements (since $x$ is fixed).
   • The number of ways to choose the remaining $k-1$ elements from $n-1$ is $\binom{n-1}{k-1}$.
   • Since any of the $k$ elements of the subset could have been chosen as $x$, there are $k$ ways to make this choice.

3. Combining these counts: Multiply the two steps above: $k \cdot \binom{n-1}{k-1}.$

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Step 4: Conclude equality

Since both sides of the equation count the same thing (the total number of subsets of $k$ elements from $n$ elements), we conclude: $\binom{n}{k} = k \cdot \binom{n-1}{k-1}.$

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Tip:

To master combinatorial proofs, focus on interpreting binomial coefficients as counting subsets and breaking problems into logical steps.

Questions:

1. Why does fixing one element of the subset simplify the proof?
2. How is the binomial coefficient $\binom{n}{k}$ defined algebraically, and how does it relate to this proof?
3. Can you derive this identity using induction instead of a combinatorial proof?
4. How does this identity relate to Pascal's triangle?
5. What happens if $k = 1$ or $k = n$ in this identity?

Would you like further details or examples?